Robust estimation with modified Huber's function for functional linear models

Abstract

In this article, we consider a new robust estimation procedure for functional linear models with both slope function and functional predictor approximated by functional principal component basis functions. A modified Huber's function with tail function substituted by the exponential squared loss (ESL) is applied to the estimation procedure for achieving robustness against outliers. The tuning parameters of the new estimation method are data-driven, which enables us to reach better robustness and efficiency than other robust methods in the presence of outliers or heavy-tailed error distribution. We will show that the resulting estimator for the slope function achieves the optimal convergence rate as the least-squares estimator does in the classical functional linear regression. The convergence rate of the prediction in terms of conditional mean squared prediction error is also established. The proposed method is illustrated with simulation studies and a real data example.